Recently (Jaekel 2023; Van Hirtum et al. 2024) the ninth Dedekind number (D(9)) was computed. In fact, the result of two independent computations, confirming each other’s result, were published nearly at the same time. In one of these, the authors of the present paper were involved. D(n) counts the monotone Boolean functions or antichains on subsets of a set of n elements. The number rises doubly exponentially in the number of elements n, and until now no algorithm of a lower combinatorial complexity is known to compute D(n). In our computation, we use coefficients representing the number of solutions of a specific set of equations on antichains over a finite set. We refer to these coefficients as P-coefficients. These can be computed efficiently. In this paper, we generalise this coefficient and apply it to four different systems of equations. Finally we show how the coefficient was used in our computation of D(9), and how its generalisations can be used to compute D(n).
